Independence of \(\ell\) and local terms

Let $k$ be an algebraically closed field and let $c:C\rightarrow X\times X$ be a correspondence. Let $\ell$ be a prime invertible in $k$ and let $K\in D^b_c(X, \overline {\mathbb Q}_\ell )$ be a complex. An action of $c$ on $K$ is by definition a map $u:c_1^*K\rightarrow c_2^!K$. For such an action one can define for each proper component $Z$ of the fixed point scheme of $c$ a local term $\text{lt}_Z(K, u)\in \overline {\mathbb Q}_\ell$. In this talk I will discuss various techniques for studying these local terms and some independence of $\ell$ results for them. I will also discuss consequences for traces of correspondences acting on cohomology.